Parametric oscillatory instability on axially-symmetrical test mass elastic modes in Advanced LIGO interferometer

We discuss the importance of accurate calculation of axially-symmetrical elastic modes frequencies and forms in test masses (mirrors) of Advanced LIGO interferometer using the method of superposition to analyse the effect of parametric oscillatory instability. This method consists in possibility to construct analytical expressions for partial solutions of media motion equation in the cylindrical coordinates which allow to satisfy all zero boundary conditions. Obtained results allow to predict more exactly (in comparison with numerical calculations) the number of combinations of optical and elastic modes that can create undesirable effect of parametric oscillatory instability.